Unit 8: Inference for Means Review

Overview

This unit focuses on inference for a population mean or the difference between two population means. It includes discussions on:

Model Properties:
- Mean of sampling distribution = population mean (µ)
- Standard deviation of sampling distribution = σ / √n
- Larger samples yield more accurate estimates due to shrinking standard error
Central Limit Theorem:
- For large enough sample size, the sampling distribution is approximately normal
- If n ≥ 30, distribution is likely normal

Confidence Interval Formula: X ± (critical value × standard error)
Types of Confidence Intervals:
- One-Sample Z Interval: Known σ
- One-Sample T Interval: Unknown σ, use sample standard deviation (s)
T Model: Used when population standard deviation is unknown
- Requires degrees of freedom (n-1 for one sample)
- Two independent samples require technology to calculate degrees of freedom
Confidence Levels & Precision:
- More precise estimates achieved with larger samples
- Wider intervals occur with higher confidence levels

Four Steps:
1. Define the hypothesis
2. Collect data (random samples)
3. Assess evidence (test statistic, P-values)
4. State conclusion (compare P-value to alpha)
Hypotheses:
- Null Hypothesis (H₀): Assumed true state of the population parameter
- Alternative Hypothesis (H₁): Reflects the research claim
Types of Tests:
- One-Tailed Test: H₁ includes µ < µ₀ or µ > µ₀
- Two-Tailed Test: H₁ includes µ ≠ µ₀

Populations should be normally distributed or sample sizes should be large enough (n ≥ 30)
Samples must be random and independent

Test Statistic (T): Different formulas based on test type
P-Value: Probability of observing a test statistic as extreme as the sample's, assuming H₀ is true
- Compare P-value to significance level (α)
- Reject H₀ if P ≤ α
- Fail to reject H₀ if P > α

Students are encouraged to apply these concepts to practice exercises, synthesizing the knowledge gained in this unit.